reserve X,Y,Z for non trivial RealBanachSpace;

theorem LOPBAN1623:
  for X,Y,Z be RealNormSpace,
      u be Point of R_NormSpace_of_BoundedLinearOperators(X,Y),
      w be Point of R_NormSpace_of_BoundedLinearOperators(Y,Z)
  holds w*(-u) = -w*u & (-w)*u = -w*u
  proof
    let X,Y,Z be RealNormSpace,
        u be Point of R_NormSpace_of_BoundedLinearOperators(X,Y),
        w be Point of R_NormSpace_of_BoundedLinearOperators(Y,Z);
    for x be Point of X holds (w*(-u)).x = (-1)*(w*u).x
    proof
      let x be Point of X;
      thus (w*(-u)).x = modetrans(w,Y,Z).(modetrans(-u,X,Y).x) by FUNCT_2:15
      .= modetrans(w,Y,Z).((-u).x ) by LOPBAN_1:def 11
      .= modetrans(w,Y,Z).(((-1)*u).x ) by RLVECT_1:16
      .= modetrans(w,Y,Z).((-1)*(u.x)) by LOPBAN_1:36
      .= (-1) * modetrans(w,Y,Z).(u.x) by LOPBAN_1:def 5
      .= (-1) * modetrans(w,Y,Z).(modetrans(u,X,Y).x) by LOPBAN_1:def 11
      .= (-1) * (w*u).x by FUNCT_2:15;
    end;
    hence w*(-u) = (-1) * (w*u) by LOPBAN_1:36
    .= -w*u by RLVECT_1:16;
    for x be Point of X holds ((-w)*u).x = (-1)*(w*u).x
    proof
      let x be Point of X;
      thus ((-w)*u).x = modetrans(-w,Y,Z).(modetrans(u,X,Y).x) by FUNCT_2:15
      .= (-w).(modetrans(u,X,Y).x ) by LOPBAN_1:def 11
      .= ((-1)*w).(modetrans(u,X,Y).x ) by RLVECT_1:16
      .= (-1)* w.(modetrans(u,X,Y).x) by LOPBAN_1:36
      .= (-1) * modetrans(w,Y,Z).(modetrans(u,X,Y).x) by LOPBAN_1:def 11
      .= (-1) * (w*u).x by FUNCT_2:15;
    end;
    hence (-w)*u = (-1) * (w*u) by LOPBAN_1:36
    .= -w*u by RLVECT_1:16;
  end;
